Surface waves. Surface acoustic waves

Waves in a discrete chain. Wave polarization. Shear wave speed. Kinetic energy density of running water.

Waves.

For a long time, the visual image of a wave has always been associated with waves on the surface of water. But water waves are a much more complex phenomenon than many other wave processes, such as the propagation of sound in a homogeneous isotropic medium. Therefore, it is natural to begin the study of wave motion not with waves on water, but with simpler cases.

Waves in a discrete chain.

The easiest way is to imagine a wave propagating along an endless chain of connected pendulums (Fig. 192). We start with an infinite chain so that we can consider a wave propagating in one direction and not think about its possible reflection from the end of the chain.

Rice. 192. Wave in a chain of connected pendulums If the pendulum, located at the beginning of the chain, is brought into harmonic oscillatory motion with a certain frequency co and amplitude A, then the oscillatory motion will propagate along the chain. This propagation of vibrations from one place to another is called a wave process or wave. In the absence of damping, any other pendulum in the chain will repeat the forced oscillations of the first pendulum with some phase lag. This delay is due to the fact that the propagation of oscillations along the chain occurs at a certain finite speed. The speed at which vibrations propagate depends on the rigidity of the spring connecting the pendulums and on how strong the connection between the pendulums is. If the first pendulum in the chain moves according to a certain law, its displacement from the equilibrium position is a given function of time, then the displacement of the pendulum, distant from the beginning of the chain by a distance, at any moment of time will be exactly the same as the displacement of the first pendulum at an earlier moment of time will be be described by a function. Let the first pendulum undergo harmonic oscillations and its displacement from the equilibrium position be given by the expression. Each of the pendulums of the chain is characterized by the distance at which it is located from the beginning of the chain. Therefore, its displacement from the equilibrium position during the passage of a wave is naturally denoted by. Then, in accordance with what was said above, we have. The wave described by the equation is called monochromatic. A characteristic feature of a monochromatic wave is that each of the pendulums performs a sinusoidal oscillation of a certain frequency. The propagation of a wave along a chain of pendulums is accompanied by a transfer of energy and momentum. But no mass transfer occurs in this case: each pendulum, oscillating around the equilibrium position, on average remains in place.

Wave polarization. Depending on the direction in which the pendulums oscillate, they speak of waves of different polarization. If the pendulums oscillate along the direction of wave propagation, as in Fig. 192, then the wave is called longitudinal, if across it is called transverse. Typically, waves of different polarization travel at different speeds. The considered chain of coupled pendulums is an example of a mechanical system with lumped parameters.

Another example of a system with lumped parameters in which waves can propagate is a chain of balls connected by light springs (Fig. 193). In such a system, inert properties are concentrated in the balls, and elastic properties in the springs. When a wave propagates, the kinetic energy of vibration is localized on the balls, and the potential energy is localized on the springs. It is easy to imagine that such a chain of balls connected by springs can be considered as a model of a one-dimensional system with distributed parameters, for example an elastic string. In a string, each element of length has both mass, inert properties, and rigidity, elastic properties. Waves in a stretched string. Let us consider a transverse monochromatic wave propagating in an infinite stretched string. Pre-tensioning of the string is necessary because an untensioned flexible string, unlike a solid rod, is elastic only with respect to tensile deformation, but not compression. A monochromatic wave in a string is described by the same expression as a wave in a chain of pendulums. However, now the role of a separate pendulum is played by each element of the string, therefore the variable in the equation characterizing the equilibrium position of the pendulum takes on continuous values. The displacement of any string element from its equilibrium position during the passage of a wave is a function of two time variables and the equilibrium position of this element. If we fix a specific string element in the formula, then the function, when fixed, gives the displacement of the selected string element depending on time. This mixing is a harmonic oscillation with frequency and amplitude. The initial phase of vibration of this element of the string depends on its equilibrium position. All elements of the string, when passing a monochromatic wave, perform harmonic vibrations of the same frequency and amplitude, but differing in phase.

Wavelength.

If we fix it in the formula and consider the entire string at the same moment in time, then the function, when fixed, gives an instantaneous picture of the displacements of all elements of the string, like an instant photograph of a wave. In this “photograph” we will see a frozen sinusoid (Fig. 194). The period of this sine wave, the distance between adjacent humps or troughs, is called the wavelength. From the formula we can find that the wavelength is related to the frequency and speed of the wave and the ratio of the oscillation period. The picture of wave propagation can be imagined if this “frozen” sinusoid is set in motion along the axis at speed.

Rice. 194. Displacement of different points of the string at the same moment in time. Rice. 195. Pictures of displacements of string points at a moment in time. Two successive “snapshots” of a wave at instants in time are shown in Fig. 195. It can be seen that the wavelength is equal to the distance traveled by any hump during the period of oscillation in accordance with the formula.

Shear wave speed.

Let us determine the speed of propagation of a monochromatic transverse wave in a string. We will assume that the amplitude is small compared to the wavelength. Let the wave run to the right with speed u. Let's move to a new frame of reference, moving along the string with a speed equal to the speed of the wave u. This reference frame is also inertial and, therefore, Newton's laws are valid in it. From this frame of reference, the wave appears to be a frozen sine wave, and the matter of the string is sliding along this sine wave to the left: any pre-colored element of the string will appear to be running away along the sine wave to the left with speed.

Rice. 196. To calculate the speed of wave propagation in a string. Let us consider in this reference frame an element of a string with a length that is much less than the wavelength at the moment when it is on the crest of the sinusoid (Fig. 196). Let us apply Newton's second law to this element. The forces acting on the element from neighboring sections of the string are shown in the highlighted circle in Fig. 196. Since a transverse wave is considered, in which the displacements of the string elements are perpendicular to the direction of propagation of the wave, then the horizontal component of the tension force. the pressure is constant along the entire string. Since the length of the section under consideration, the directions of the tension forces acting on the selected element are almost horizontal, and their modulus can be considered equal. The resultant of these forces is directed downward and equal. The speed of the element under consideration is equal to and directed to the left, and a small section of its sinusoidal trajectory near the hump can be considered an arc of a circle of radius. Therefore, the acceleration of this string element is downward and equal. The mass of a string element can be represented as the density of the string material, and the cross-sectional area, which, due to the smallness of the deformations during wave propagation, can be considered the same as in the absence of a wave. Based on Newton's second law. This is the desired speed of propagation of a transverse monochromatic wave of small amplitude in a stretched string. It can be seen that it depends only on the mechanical stress of the stretched string and its density and does not depend on the amplitude and wavelength. This means that transverse waves of any length propagate in a stretched string at the same speed. If, for example, two monochromatic waves with identical amplitudes and similar frequencies co propagate simultaneously in a string, then “instant photographs” of these monochromatic waves and the resulting wave will have the form shown in Fig. 197.

Where the hump of one wave coincides with the hump of another, the mixing in the resulting wave is maximum. Since the sinusoids corresponding to the individual waves run along the z axis at the same speed and, the resulting curve runs at the same speed without changing its shape. It turns out that this is true for a wave disturbance of any shape: transverse waves of any type propagate in a stretched string without changing their shape. About wave dispersion. If the speed of propagation of monochromatic waves does not depend on wavelength or frequency, then they say that there is no dispersion. The preservation of the shape of any wave during its propagation is a consequence of the absence of dispersion. There is no dispersion for waves of any type propagating in continuous elastic media. This circumstance makes it very easy to find the speed of longitudinal waves.

Velocity of longitudinal waves.

Let us consider, for example, a long elastic rod of area in which a longitudinal disturbance with a steep leading edge propagates. Let at some point in time this front, moving with speed, reach a point with a coordinate to the right of the front; all points of the rod are still at rest. After a period of time, the front will move to the right by a distance (Fig. 198). Within this layer, all particles move at the same speed. After this period of time, the particles of the rod, which were at the wave front at the moment, will move along the rod a distance. Let us apply the law of conservation of momentum to the mass of the rod involved in the wave process over time. Let us express the force acting on the mass through the deformation of the rod element using Hooke's law. The length of the selected element of the rod is equal, and the change in its length under the action of force is equal. Therefore, with the help of we find Substituting this value in, we obtain The speed of longitudinal sound waves in an elastic rod depends only on Young’s modulus and density. It is easy to see that in most metals this speed is approximately. The speed of longitudinal waves in an elastic medium is always greater than the speed of transverse waves. Let us compare, for example, the velocities of longitudinal and transverse waves u(in a stretched flexible string. Since at small deformations the elastic constants do not depend on the applied forces, the velocity of longitudinal waves in a stretched string does not depend on its pretension and is determined by the formula. In order to compare this speed with the previously found speed of transverse waves u we express the tension force of the string included in the formula through the relative deformation of the string due to this pre-tension. Substituting the value into the formula, we obtain Thus, the speed of transverse waves in a tense string ut turns out to be significantly less than the speed of longitudinal waves, so as the relative stretching of the string e is much less than unity. Wave energy. When waves propagate, energy is transferred without transfer of matter. The energy of a wave in an elastic medium consists of the kinetic energy of oscillating particles of the substance and the potential energy of elastic deformation of the medium. Consider, for example, a longitudinal wave in elastic rod. At a fixed moment in time, kinetic energy is distributed unevenly throughout the volume of the rod, since some points of the rod are at rest at this moment, while others, on the contrary, are moving at maximum speed. The same is true for potential energy, since at this moment some elements of the rod are not deformed, while others are deformed to the maximum. Therefore, when considering wave energy, it is natural to introduce the density of kinetic and potential energies. The wave energy density at each point of the medium does not remain constant, but changes periodically as the wave passes: the energy spreads along with the wave.

Why, when a transverse wave propagates in a stretched string, is the longitudinal component of the string tension force the same along the entire string and does not change as the wave passes?

What are monochromatic waves? How is the length of a monochromatic wave related to frequency and speed of propagation? In what cases are waves called longitudinal and in what cases are they called transverse? Show using qualitative reasoning that the speed of wave propagation is greater, the greater the force tending to return the disturbed section of the medium to a state of equilibrium, and the less, the greater the inertia of this section. What characteristics of the medium determine the speed of longitudinal waves and the speed of transverse waves? How are the velocities of such waves in a stretched string related to each other?

Kinetic energy density of a traveling wave.

Let us consider the kinetic energy density in a monochromatic elastic wave described by the equation. Let us select a small element in the rod between the planes such that its length in the undeformed state is much less than the wavelength. Then the velocities of all particles of the rod in this element during wave propagation can be considered the same. Using the formula, we find the speed, considering it as a function of time and considering the value characterizing the position of the rod element in question to be fixed. The mass of the selected element of the rod, therefore its kinetic energy at the moment of time is Using the expression, we find the density of kinetic energy at the point at the moment of time. Potential energy density. Let's move on to calculating the potential energy density of the wave. Since the length of the selected element of the rod is small compared to the length of the wave, the deformation of this element caused by the wave can be considered homogeneous. Therefore, the potential strain energy can be written as the elongation of the rod element under consideration caused by a passing wave. To find this extension, you need to consider the position of the planes limiting the selected element at some point in time. The instantaneous position of any plane, the equilibrium position of which is characterized by a coordinate, is determined by a function considered as a function at a fixed. Therefore, the elongation of the rod element under consideration, as can be seen from Fig. 199, is equal to The relative elongation of this element is If in this expression we go to the limit at, then it turns into the derivative of the function with respect to the variable at fixed. Using the formula we get

Rice. 199. To calculate the relative elongation of the rod Now the expression for potential energy takes the form and the density of potential energy at a point at an instant of time is the Energy of the traveling wave. Since the speed of propagation of longitudinal waves, the right-hand sides in the formulas coincide. This means that in a traveling longitudinal elastic wave the densities of kinetic and potential energies are equal at any moment of time at any point in the medium. The dependence of the wave energy density on the coordinate at a fixed time is shown in Fig. 200. Let us note that, in contrast to localized oscillations (oscillator), where kinetic and potential energies change in antiphase, in a traveling wave oscillations of kinetic and potential energies occur in the same phase. Kinetic and potential energies at each point in the medium simultaneously reach maximum values and simultaneously become zero. The equality of the instantaneous values of the density of kinetic and potential energies is a general property of traveling waves of waves propagating in a certain direction. It can be seen that this is also true for transverse waves in a stretched flexible string. Rice. 200. Displacement of particles of the medium and energy density in a traveling wave

Until now, we have considered waves propagating in a system that has infinite extension in only one direction: in a chain of pendulums, in a string, in a rod. But waves can also propagate in a medium that has infinite dimensions in all directions. In such a continuous medium, waves come in different types depending on the method of their excitation. Plane wave. If, for example, a wave arises as a result of harmonic oscillations of an infinite plane, then in a homogeneous medium it propagates in a direction perpendicular to this plane. In such a wave, the displacement of all points of the medium lying on any plane perpendicular to the direction of propagation occurs in exactly the same way. If wave energy is not absorbed in the medium, then the amplitude of oscillations of points in the medium is the same everywhere and their displacement is given by the formula. Such a wave is called a plane wave.

Spherical wave.

A different type of spherical wave is created in a homogeneous isotropic elastic medium by a pulsating ball. Such a wave propagates at the same speed in all directions. Its wave surfaces, surfaces of constant phase, are concentric spheres. In the absence of energy absorption in the medium, it is easy to determine the dependence of the amplitude of a spherical wave on the distance to the center. Since the flow of wave energy, proportional to the square of the amplitude, is the same through any sphere, the amplitude of the wave decreases in inverse proportion to the distance from the center. The equation of a longitudinal spherical wave has the form where is the amplitude of oscillations at a distance from the center of the wave.

How does the energy transferred by a traveling wave depend on the frequency and amplitude of the wave?

What is a plane wave? Spherical wave? How do the amplitudes of plane and spherical waves depend on distance?

Explain why in a traveling wave the kinetic energy and potential energy change in the same phase.

Elastic waves propagating along the free boundary of a solid or along the boundary of a solid with other media

Animation

Description

The existence of surface waves (SW) is a consequence of the interaction of longitudinal and (or) transverse elastic waves when these waves are reflected from a flat boundary between different media under certain boundary conditions for the displacement components. PVs in solids are of two classes: with vertical polarization, in which the vector of vibrational displacement of particles of the medium is located in a plane perpendicular to the boundary surface, and with horizontal polarization, in which the vector of displacement of particles of the medium is parallel to the boundary surface.

The most common special cases of PV include the following.

1) Rayleigh waves (or Rayleigh waves), propagating along the boundary of a solid body with a vacuum or a fairly rarefied gaseous medium. The energy of these waves is localized in a surface layer with a thickness of l to 2l, where l is the wavelength. Particles in a Rayleigh wave move along ellipses, the major semi-axis w of which is perpendicular to the boundary, and the minor semi-axis u is parallel to the direction of propagation of the wave (Fig. 1a).

Surface elastic Rayleigh wave on the free boundary of a solid body

Rice. 1a

Designations:

The phase velocity of Rayleigh waves is c R » 0.9c t , where c t is the phase velocity of a plane transverse wave.

2) Damped waves of the Rayleigh type at the boundary of a solid body with a liquid, provided that the phase velocity in the liquid with L< с R в твердом теле (что справедливо почти для всех реальных сред). Эта волна непрерывно излучает энергию в жидкость, образуя в ней отходящую от границы неоднородную волну (рис. 1б).

Surface elastic damped wave of Rayleigh type at the boundary of a solid body and a liquid

Rice. 1b

Designations:

x is the direction of wave propagation;

u,w - particle displacement components;

the curves depict the progression of changes in the amplitude of displacements with distance from the boundary;

inclined lines are the fronts of the outgoing wave.

The phase velocity of this wave is equal to R, up to a percentage, and the attenuation coefficient at wavelength al ~ 0.1. The depth distribution of displacements and stresses is the same as in the Rayleigh wave.

3) A continuous wave with vertical polarization, traveling along the boundary of a liquid and a solid with a speed less than L (and, accordingly, less than the speeds of longitudinal and transverse waves in a solid). The structure of this PV is completely different from that of the Rayleigh wave. It consists of a weakly inhomogeneous wave in a liquid, the amplitude of which slowly decreases with distance from the boundary, and two strongly inhomogeneous longitudinal and transverse waves in a solid (Fig. 1c).

Undamped PV at the solid-liquid interface

Rice. 1c

Designations:

x is the direction of wave propagation;

u,w - particle displacement components;

the curves depict the progression of changes in the displacement amplitude with distance from the boundary.

The energy of the wave and the movement of particles are localized mainly in the liquid.

4) A Stoneley wave propagating along a flat boundary of two solid media whose elastic moduli and densities do not differ much. Such a wave consists (Fig. 1d) as if of two Rayleigh waves - one in each medium.

Surface elastic Stonley wave at the interface of two solid media

Rice. 1g

Designations:

x is the direction of wave propagation;

u,w - particle displacement components;

the curves depict the progression of changes in the displacement amplitude with distance from the boundary.

The vertical and horizontal components of the displacements in each medium decrease with distance from the boundary so that the wave energy is concentrated in two boundary layers of thickness ~l. The phase velocity of the Stoneley wave is less than the values of the phase velocities of longitudinal and transverse waves in both adjacent media.

5) Love waves - SW with horizontal polarization, which can propagate at the boundary of a solid half-space with a solid layer (Fig. 1e).

Surface elastic Love wave at the boundary “solid half-space - solid layer”

Rice. 1d

Designations:

x is the direction of wave propagation;

the curves depict the progression of changes in the displacement amplitude with distance from the boundary.

These waves are purely transverse: they have only one displacement component v, and the elastic deformation in a Love wave is pure shear. Displacements in the layer (index 1) and in the half-space (index 2) are described by the expressions:

v 1 = (A ¤ cos(s 1 h)) cos(s 1 (h - z))sin(w t - kx) ;

v 2 = A H exp(s 2 z) sin(w t - kx ),

where t is time;

w - circular frequency;

s 1 = ( k t1 2 - k 2 )1/2 ;

s 2 = (k 2 - k t2 2 )1/2;

k is the wave number of the Love wave;

k t1, k t2 - wave numbers of transverse waves in the layer and in the half-space, respectively;

h - layer thickness;

A is an arbitrary constant.

From the expressions for v 1 and v 2 it is clear that the displacements in the layer are distributed along the cosine, and in the half-space they decrease exponentially with depth. Love waves are characterized by velocity dispersion. At small layer thicknesses, the phase velocity of the Love wave tends to the phase velocity of the bulk transverse wave in the half-space. For w h ¤ c t2 >>1, Love waves exist in the form of several modifications, each of which corresponds to a normal wave of a certain order.

Waves on the free surface of a liquid or at the interface between two immiscible liquids are also considered wave waves. Such PVs arise under the influence of external influences, for example, wind, which removes the surface of the liquid from an equilibrium state. In this case, however, elastic waves cannot exist. Depending on the nature of the restoring forces, 3 types of PVs are distinguished: gravitational, caused mainly by gravity; capillary, caused mainly by surface tension forces; gravity-capillary (see description of FE “Surface waves in liquid”).

Timing characteristics

Initiation time (log to -3 to -1);

Lifetime (log tc from -1 to 3);

Degradation time (log td from -1 to 1);

Optimal development time (log tk from 0 to 1).

Diagram:

Technical implementations of the effect

Technical implementation of the effect

A Rayleigh wave can be obtained on the free surface of a sufficiently extended solid body (solid-air boundary). To do this, the emitter of elastic waves (longitudinal, transverse) is placed on the surface of the body (Fig. 2), although, in principle, the source of the waves can also be located inside the medium at some depth (earthquake source model).

Generation of a Rayleigh wave at the free boundary of a solid body

Rice. 2

Applying an effect

Since seismic PVs weakly attenuate with distance, PVs, primarily Rayleigh and Love, are used in geophysics to determine the structure of the earth's crust. In ultrasonic flaw detection, PV is used for comprehensive non-destructive testing of the surface and surface layer of a sample. In acoustoelectronics (AE), using PV, it is possible to create microelectronic circuits for processing electrical signals. The advantages of PV in AE devices are low conversion losses during excitation and reception of PV, the availability of the wave front, which allows you to pick up a signal and control the propagation of the wave at any point in the sound pipeline, etc.

Example of AE devices on PV: resonator (Fig. 3).

Resonance structure on surface acoustic waves

Rice. 3

Designations:

1 - converter;

2 - reflector system (metal electrodes or grooves).

Quality factor up to 104, low losses (less than 5 dB), frequency range 30 - 1000 MHz. Operating principle. A standing PV is created between the reflectors 2, which is generated and received by the converter 1.

Literature

1. Ultrasound / Ed. I.P. Golyamina.- M.: Soviet Encyclopedia, 1979.- P. 400.

2. Brekhovskikh L.M., Goncharov V.V. Introduction to continuum mechanics. - M.: Nauka, 1982.

Keywords

amplitude
surface wave
Rayleigh wave
Love wave
Stonley wave
vertically polarized wave
horizontally polarized wave
wavelength
wave speed
velocity dispersion
frequency

Sections of natural sciences:

Surface acoustic waves(SAW) - elastic waves propagating along the surface of a solid body or along the boundary with other media. Surfactants are divided into two types: with vertical polarization and with horizontal polarization ( Love waves).

The most common special cases of surface waves include the following:

Rayleigh waves(or Rayleigh), in the classical sense, propagating along the boundary of an elastic half-space with a vacuum or a fairly rarefied gaseous medium.
at the solid-liquid interface.
, running along the boundary of a liquid and a solid body
Stoneleigh Wave, propagating along the flat boundary of two solid media, the elastic moduli and density of which do not differ much.
Love waves- surface waves with horizontal polarization (SH type), which can propagate in the elastic layer structure on an elastic half-space.

Encyclopedic YouTube

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✪ Seismic waves

✪ Longitudinal and transverse waves. Sound waves. Lesson 120

✪ Lecture seven: Waves

Subtitles

In this video I want to discuss seismic waves a little bit. Let's write down the topic. Firstly, they are very interesting in themselves and, secondly, they are very important for understanding the structure of the Earth. You have already seen my video about the layers of the Earth, and it was thanks to seismic waves that we concluded what layers our planet consists of. And while seismic waves are usually associated with earthquakes, they are actually any waves that travel along the ground. They can come from an earthquake, a strong explosion, anything that can send a lot of energy directly into the ground and stone. So, there are two main types of seismic waves. And we'll focus more on one of them. The first is surface waves. Let's write it down. The second is body waves. Surface waves are simply waves that travel across the surface of something. In our case, on the surface of the earth. Here, in the illustration, you can see what surface waves look like. They are similar to the ripples that can be seen on the surface of water. There are two types of surface waves: Rayleigh waves and Love waves. I won't go into detail, but here you can see that Rayleigh waves move up and down. This is where the earth moves up and down. It's moving down here. Here it's up. And then - down again. It looks like a wave running across the earth. Love waves, in turn, move sideways. That is, here the wave does not move up and down, but, if you look in the direction of the wave, it moves to the left. Here it moves to the right. Here - to the left. Here - again to the right. In both cases, the movement of the wave is perpendicular to the direction of its movement. Sometimes such waves are called transverse waves. And they, as I said, are like waves in water. Body waves are much more interesting because, firstly, they are the fastest waves. And, besides, it is these waves that are used to study the structure of the earth. Body waves come in two types. There are P-waves, or primary waves. And S-waves, or secondary. They can be seen here. Such waves are energy moving inside the body. And not just on its surface. So, in this picture, which I downloaded from Wikipedia, you can see how a large stone is being hit with a hammer. And when the hammer hits the stone... Let me redraw it larger. Here I will have a stone and I will hit it with a hammer. It will compress the stone where it hits. Then the energy from the impact will push the molecules, which will crash into the molecules next door. And these molecules will crash into the molecules behind them, and those, in turn, into the molecules next to them. It turns out that this compressed part of the stone moves as a wave. These are compressed molecules, they will crash into the molecules nearby and then the stone here will become denser. The first molecules, the ones that started the whole movement, will return to their place. Therefore, the compression has moved, and will move further. The result is a compression wave. You hit this with a hammer and you get a changing density that moves in the direction of the wave. In our case, the molecules move back and forth along the same axis. Parallel to the direction of the wave. These are P-waves. P waves can travel in air. Essentially, sound waves are compression waves. They can move in both liquids and solids. And, depending on the environment, they move at different speeds. In the air they move at a speed of 330 m/s, which is not so slow for everyday life. In liquid they move at a speed of 1,500 m/s. And in granite, which makes up most of the Earth's surface, they move at a speed of 5,000 m/s. Let me write this down. 5,000 meters, or 5 km/s in granite. And I'll draw the S-waves now, because this one is too small. If you hit this area with a hammer, the force of the impact will temporarily move the stone to the side. It will be slightly deformed and will pull the adjacent section of stone along with it. This rock on top will then be pulled down, and the rock that was originally hit will return up. And after about a millisecond, the layer of stone on top deforms slightly to the right. And then, over time, the deformation will move upward. Notice that in this case the wave is also moving upward. But the movement of the material is no longer parallel to the axis, as in P-waves, but perpendicular. These perpendicular waves are also called transverse vibrations. The movement of particles is perpendicular to the axis of wave movement. These are S-waves. They move a little slower than P-waves. Therefore, if there is an earthquake, you will first feel the P waves. And then, at approximately 60% of the speed of P-waves, S-waves will come. So, to understand the structure of the Earth, it is important to remember that S waves can only move in solid matter. Let's write this down. You could say that you saw transverse waves on the water. But there were surface waves. And we are discussing body waves. Waves that travel within a volume of water. To make it easier to imagine, I'll draw some water, let's say there will be a pool here. In the context. Something like that. Yes, I could have drawn it better. So here's a cutaway view of the pool, and I hope you can understand what's going on in it. And if I compress some of the water, for example, by hitting it with something very large, the water will quickly compress. The P-wave will be able to move because the water molecules will crash into the molecules next to them, which will crash into the molecules behind them. And this compression, this P-wave, will move in the direction of my impact. This shows that the P-wave can move both in liquids and, for example, in air. Fine. And remember that we are talking about underwater waves. Not about surfaces. Our waves move in the volume of water. Let's assume that we took a hammer and hit a given volume of water from the side. And this will only create a wave of compression in this direction. And nothing more. A transverse wave will not arise because the wave does not have the elasticity that allows its parts to oscillate from side to side. The S-wave requires the kind of elasticity that only occurs in solids. In what follows, we will use the properties of P waves, which can travel in air, liquids, and solids, and the properties of S waves to find out what the earth is made of. Subtitles by the Amara.org community

Rayleigh waves

Damped Rayleigh waves

Damped Rayleigh-type waves at the solid-liquid interface.

Continuous wave with vertical polarization

Continuous wave with vertical polarization, running along the boundary of a liquid and a solid at the speed of sound in a given medium.

Wave(Wave, surge, sea) - formed due to the adhesion of particles of liquid and air; sliding along the smooth surface of the water, at first the air creates ripples, and only then, acting on its inclined surfaces, gradually develops agitation of the water mass. Experience has shown that water particles do not have forward motion; moves only vertically. Sea waves are the movement of water on the sea surface that occurs at certain intervals.

The highest point of the wave is called comb or the top of the wave, and the lowest point is sole. Height of a wave is the distance from the crest to its base, and length this is the distance between two ridges or soles. The time between two crests or troughs is called period waves.

Main causes

On average, the height of a wave during a storm in the ocean reaches 7-8 meters, usually it can stretch in length - up to 150 meters and up to 250 meters during a storm.

In most cases, sea waves are formed by the wind. The strength and size of such waves depend on the strength of the wind, as well as its duration and “acceleration” - the length of the path along which the wind acts on the water surface. Sometimes the waves that hit the coast can originate thousands of kilometers from the coast. But there are many other factors in the occurrence of sea waves: these are the tidal forces of the Moon and the Sun, fluctuations in atmospheric pressure, eruptions of underwater volcanoes, underwater earthquakes, and the movement of sea vessels.

Waves observed in other water bodies can be of two types:

1) Wind created by the wind, taking on a steady character after the wind ceases to act and called established waves, or swell; Wind waves are created due to the action of wind (movement of air masses) on the surface of the water, that is, injection. The reason for the oscillatory movements of the waves becomes easy to understand if you notice the effect of the same wind on the surface of a wheat field. The inconstancy of wind flows, which create waves, is clearly visible.

2) Waves of movement, or standing waves, are formed as a result of strong tremors at the bottom during earthquakes or excited, for example, by a sharp change in atmospheric pressure. These waves are also called single waves.

Unlike tides and currents, waves do not move masses of water. The waves move, but the water remains in place. A boat that rocks on the waves does not float away with the wave. She will be able to move slightly along an inclined slope only thanks to the force of earth's gravity. Water particles in a wave move along rings. The further these rings are from the surface, the smaller they become and, finally, disappear completely. Being in a submarine at a depth of 70-80 meters, you will not feel the effect of sea waves even during the most severe storm on the surface.

Types of sea waves

Waves can travel vast distances without changing shape and losing virtually no energy, long after the wind that caused them has died down. Breaking on the shore, sea waves release enormous energy accumulated during the journey. The force of continuously breaking waves changes the shape of the shore in different ways. The spreading and rolling waves wash the shore and are therefore called constructive. Waves crashing onto the shore gradually destroy it and wash away the beaches that protect it. That's why they are called destructive.

Low, wide, rounded waves away from the shore are called swells. Waves cause water particles to describe circles and rings. The size of the rings decreases with depth. As the wave approaches the sloping shore, the water particles in it describe increasingly flattened ovals. Approaching the shore, the sea waves can no longer close their ovals, and the wave breaks. In shallow water, the water particles can no longer close their ovals, and the wave breaks. Headlands are formed from harder rock and erode more slowly than adjacent sections of the coast. Steep, high sea waves undermine the rocky cliffs at the base, creating niches. Cliffs sometimes collapse. The terrace, smoothed by the waves, is all that remains of the rocks destroyed by the sea. Sometimes water rises along vertical cracks in the rock to the top and breaks out to the surface, forming a funnel. The destructive force of the waves widens the cracks in the rock, forming caves. When the waves wear away at the rock on both sides until they meet at a break, arches are formed. When the top of the arch falls into the sea, stone pillars remain. Their foundations are undermined and the pillars collapse, forming boulders. The pebbles and sand on the beach are the result of erosion.

Destructive waves gradually erode the coast and carry away sand and pebbles from sea beaches. Bringing the full weight of their water and washed-away material onto slopes and cliffs, the waves destroy their surface. They squeeze water and air into every crack, every crevice, often with explosive energy, gradually separating and weakening the rocks. The broken rock fragments are used for further destruction. Even the hardest rocks are gradually destroyed, and the land on the shore changes under the influence of waves. Waves can destroy the seashore with amazing speed. In Lincolnshire, England, erosion (destruction) is advancing at a rate of 2 m per year. Since 1870, when the largest lighthouse in the United States was built at Cape Hatteras, the sea has washed away beaches 426 m inland.

Tsunami

Tsunami These are waves of enormous destructive power. They are caused by underwater earthquakes or volcanic eruptions and can cross oceans faster than a jet plane: 1000 km/h. In deep waters, they can be less than one meter, but, approaching the shore, they slow down and grow to 30-50 meters before collapsing, flooding the shore and sweeping away everything in their path. 90% of all recorded tsunamis occurred in the Pacific Ocean.

The most common reasons.

About 80% of tsunami generation cases are underwater earthquakes. During an earthquake under water, a mutual vertical displacement of the bottom occurs: part of the bottom sinks, and part rises. Oscillatory movements occur vertically on the surface of the water, tending to return to the original level - the average sea level - and generate a series of waves. Not every underwater earthquake is accompanied by a tsunami. Tsunamigenic (that is, generating a tsunami wave) is usually an earthquake with a shallow source. The problem of recognizing the tsunamigenicity of an earthquake has not yet been solved, and warning services are guided by the magnitude of the earthquake. The most powerful tsunamis are generated in subduction zones. Also, it is necessary for the underwater shock to resonate with the wave oscillations.

Landslides. Tsunamis of this type occur more frequently than estimated in the 20th century (about 7% of all tsunamis). Often an earthquake causes a landslide and it also generates a wave. On July 9, 1958, an earthquake in Alaska caused a landslide in Lituya Bay. A mass of ice and earth rocks collapsed from a height of 1100 m. A wave was formed that reached a height of more than 524 m on the opposite shore of the bay. Cases of this kind are quite rare and are not considered as a standard. But underwater landslides occur much more often in river deltas, which are no less dangerous. An earthquake can cause a landslide and, for example, in Indonesia, where shelf sedimentation is very large, landslide tsunamis are especially dangerous, as they occur regularly, causing local waves more than 20 meters high.

Volcanic eruptions account for approximately 5% of all tsunami events. Large underwater eruptions have the same effect as earthquakes. In large volcanic explosions, not only are waves generated from the explosion, but water also fills the cavities of the erupted material or even the caldera, resulting in a long wave. A classic example is the tsunami generated after the Krakatoa eruption in 1883. Huge tsunamis from the Krakatoa volcano were observed in harbors around the world and destroyed a total of more than 5,000 ships and killed about 36,000 people.

Signs of a tsunami.

Sudden fast the withdrawal of water from the shore over a considerable distance and the drying of the bottom. The further the sea recedes, the higher the tsunami waves can be. People who are on the shore and do not know about dangers, may stay out of curiosity or to collect fish and shells. In this case, it is necessary to leave the shore as soon as possible and move as far away from it as possible - this rule should be followed when, for example, in Japan, on the Indian Ocean coast of Indonesia, or Kamchatka. In the case of a teletsunami, the wave usually approaches without the water receding.
Earthquake. The epicenter of an earthquake is usually in the ocean. On the coast, the earthquake is usually much weaker, and often there is no earthquake at all. In tsunami-prone regions, there is a rule that if an earthquake is felt, it is better to move further from the coast and at the same time climb a hill, thus preparing in advance for the arrival of the wave.
Unusual drift ice and other floating objects, formation of cracks in fast ice.
Huge reverse faults at the edges of stationary ice and reefs, the formation of crowds and currents.

rogue waves

rogue waves(Roaming waves, monster waves, freak waves - anomalous waves) - giant waves that arise in the ocean, more than 30 meters high, have behavior unusual for sea waves.

Just 10-15 years ago, scientists considered sailors’ stories about gigantic killer waves that appear out of nowhere and sink ships as just maritime folklore. For a long time wandering waves were considered fiction, since they did not fit into any mathematical model that existed at that time for calculating the occurrence and their behavior, because waves with a height of more than 21 meters cannot exist in the oceans of planet Earth.

One of the first descriptions of a monster wave dates back to 1826. Its height was more than 25 meters and it was noticed in the Atlantic Ocean near the Bay of Biscay. Nobody believed this message. And in 1840, the navigator Dumont d'Urville risked appearing at a meeting of the French Geographical Society and declaring that he had seen a 35-meter wave with his own eyes. Those present laughed at him. But there are stories about huge ghost waves that suddenly appeared in the middle of the ocean even with little storm, and their steepness resembled sheer walls of water, it became more and more.

Historical evidence of rogue waves

So, in 1933, the US Navy ship Ramapo was caught in a storm in the Pacific Ocean. For seven days the ship was tossed about by the waves. And on the morning of February 7, a shaft of incredible height suddenly crept up from behind. First, the ship was thrown into a deep abyss, and then lifted almost vertically onto a mountain of foaming water. The crew, who were lucky enough to survive, recorded a wave height of 34 meters. It moved at a speed of 23 m/sec, or 85 km/h. So far, this is considered the highest rogue wave ever measured.

During World War II, in 1942, the Queen Mary liner carried 16 thousand American military personnel from New York to the UK (by the way, a record for the number of people transported on one ship). Suddenly a 28-meter wave appeared. “The upper deck was at its usual height, and suddenly - suddenly! - it suddenly went down,” recalled Dr. Norval Carter, who was on board the ill-fated ship. The ship tilted at an angle of 53 degrees - if the angle had been even three degrees more, death would have been inevitable. The story of "Queen Mary" formed the basis of the Hollywood film "Poseidon".

However, on January 1, 1995, on the Dropner oil platform in the North Sea off the coast of Norway, a wave with a height of 25.6 meters, called the Dropner wave, was first recorded by instruments. The Maximum Wave project allowed us to take a fresh look at the causes of the death of dry cargo ships that transported containers and other important cargo. Further research recorded over three weeks around the globe more than 10 single giant waves, the height of which exceeded 20 meters. The new project is called Wave Atlas, which provides for the compilation of a worldwide map of observed monster waves and its subsequent processing and addition.

Causes

There are several hypotheses about the causes of extreme waves. Many of them lack common sense. The simplest explanations are based on the analysis of a simple superposition of waves of different lengths. Estimates, however, show that the probability of extreme waves in such a scheme is too small. Another noteworthy hypothesis suggests the possibility of focusing wave energy in some surface current structures. These structures, however, are too specific for an energy focusing mechanism to explain the systematic occurrence of extreme waves. The most reliable explanation for the occurrence of extreme waves should be based on the internal mechanisms of nonlinear surface waves without involving external factors.

Interestingly, such waves can be both crests and troughs, which is confirmed by eyewitnesses. Further research involves the effects of nonlinearity in wind waves, which can lead to the formation of small groups of waves (packets) or individual waves (solitons) that can travel long distances without significantly changing their structure. Similar packages have also been observed many times in practice. The characteristic features of such groups of waves, confirming this theory, are that they move independently of other waves and have a small width (less than 1 km), with heights decreasing sharply at the edges.

However, it has not yet been possible to completely clarify the nature of the anomalous waves.

2. Mechanical wave.

3. Source of mechanical waves.

4. Point source of waves.

5. Transverse wave.

6. Longitudinal wave.

7. Wave front.

9. Periodic waves.

10. Harmonic wave.

11. Wavelength.

12. Speed of spread.

13. Dependence of wave speed on the properties of the medium.

14. Huygens' principle.

15. Reflection and refraction of waves.

16. Law of wave reflection.

17. The law of wave refraction.

18. Plane wave equation.

19. Wave energy and intensity.

20. The principle of superposition.

21. Coherent oscillations.

22. Coherent waves.

23. Interference of waves. a) condition of interference maximum, b) condition of interference minimum.

24. Interference and the law of conservation of energy.

25. Wave diffraction.

26. Huygens–Fresnel principle.

27. Polarized wave.

29. Sound volume.

30. Pitch of sound.

31. Timbre of sound.

32. Ultrasound.

33. Infrasound.

34. Doppler effect.

1.Wave - This is the process of propagation of vibrations of any physical quantity in space. For example, sound waves in gases or liquids represent the propagation of pressure and density fluctuations in these media. An electromagnetic wave is the process of propagation of oscillations in the strength of electric magnetic fields in space.

Energy and momentum can be transferred in space by transfer of matter. Any moving body has kinetic energy. Therefore, it transfers kinetic energy by transporting matter. The same body, being heated, moving in space transfers thermal energy, transferring matter.

Particles of an elastic medium are interconnected. Disturbances, i.e. deviations from the equilibrium position of one particle are transmitted to neighboring particles, i.e. energy and momentum are transferred from one particle to neighboring particles, while each particle remains near its equilibrium position. Thus, energy and momentum are transferred along a chain from one particle to another and no transfer of matter occurs.

So, the wave process is a process of transfer of energy and momentum in space without transfer of matter.

2. Mechanical wave or elastic wave– disturbance (oscillation) propagating in an elastic medium. The elastic medium in which mechanical waves propagate is air, water, wood, metals and other elastic substances. Elastic waves are called sound waves.

3. Source of mechanical waves- a body that performs an oscillatory movement while in an elastic medium, for example, vibrating tuning forks, strings, vocal cords.

4. Point wave source – a wave source whose size can be neglected compared to the distance over which the wave travels.

5. Transverse wave – a wave in which particles of the medium oscillate in a direction perpendicular to the direction of propagation of the wave. For example, waves on the surface of water are transverse waves, because vibrations of water particles occur in a direction perpendicular to the direction of the water surface, and the wave propagates along the surface of the water. A transverse wave propagates along a cord, one end of which is fixed, the other oscillates in the vertical plane.

A transverse wave can propagate only along the interface between different media.

6. Longitudinal wave – a wave in which oscillations occur in the direction of propagation of the wave. A longitudinal wave occurs in a long helical spring if one end is subjected to periodic disturbances directed along the spring. An elastic wave running along a spring represents a propagating sequence of compression and extension (Fig. 88)

A longitudinal wave can propagate only inside an elastic medium, for example, in air, in water. In solids and liquids, both transverse and longitudinal waves can propagate simultaneously, because a solid and a liquid are always limited by a surface - the interface between two media. For example, if a steel rod is hit at the end with a hammer, then elastic deformation will begin to spread in it. A transverse wave will run along the surface of the rod, and a longitudinal wave (compression and rarefaction of the medium) will propagate inside it (Fig. 89).

7. Wave front (wave surface)– the geometric locus of points oscillating in the same phases. On the wave surface, the phases of the oscillating points at the moment in time under consideration have the same value. If you throw a stone into a calm lake, then transverse waves in the form of a circle will begin to spread across the surface of the lake from the place where it fell, with the center at the place where the stone fell. In this example, the wave front is a circle.

In a spherical wave, the wave front is a sphere. Such waves are generated by point sources.

At very large distances from the source, the curvature of the front can be neglected and the wave front can be considered flat. In this case, the wave is called plane.

8. Beam – straight line normal to the wave surface. In a spherical wave, the rays are directed along the radii of the spheres from the center, where the source of the waves is located (Fig. 90).

In a plane wave, the rays are directed perpendicular to the front surface (Fig. 91).

9. Periodic waves. When talking about waves, we meant a single disturbance propagating in space.

If the source of the waves performs continuous oscillations, then elastic waves traveling one after another appear in the medium. Such waves are called periodic.

10. Harmonic wave– a wave generated by harmonic oscillations. If a wave source performs harmonic oscillations, then it generates harmonic waves - waves in which particles vibrate according to a harmonic law.

11. Wavelength. Let a harmonic wave propagate along the OX axis, and oscillations in it occur in the direction of the OY axis. This wave is transverse and can be depicted as a sine wave (Fig. 92).

Such a wave can be obtained by causing vibrations in the vertical plane of the free end of the cord.

Wavelength is the distance between two nearest points A and B, oscillating in the same phases (Fig. 92).

12. Wave propagation speed– a physical quantity numerically equal to the speed of propagation of vibrations in space. From Fig. 92 it follows that the time during which the oscillation propagates from point to point A to the point IN, i.e. at a distance the wavelength is equal to the oscillation period. Therefore, the speed of wave propagation is equal to

13. Dependence of the speed of wave propagation on the properties of the medium. The frequency of oscillations when a wave occurs depends only on the properties of the wave source and does not depend on the properties of the medium. The speed of wave propagation depends on the properties of the medium. Therefore, the wavelength changes when crossing the interface between two different media. The speed of the wave depends on the connection between the atoms and molecules of the medium. The bond between atoms and molecules in liquids and solids is much tighter than in gases. Therefore, the speed of sound waves in liquids and solids is much greater than in gases. In air, the speed of sound under normal conditions is 340, in water 1500, and in steel 6000.

The average speed of thermal motion of molecules in gases decreases with decreasing temperature and, as a result, the speed of wave propagation in gases decreases. In a denser, and therefore more inert, medium, the wave speed is lower. If sound travels in air, its speed depends on the density of the air. Where the air density is greater, the speed of sound is less. And vice versa, where the air density is less, the speed of sound is greater. As a result, when sound propagates, the wave front is distorted. Above a swamp or above a lake, especially in the evening, the air density near the surface due to water vapor is greater than at a certain height. Therefore, the speed of sound near the surface of the water is less than at a certain height. As a result, the wave front turns in such a way that the upper part of the front bends more and more towards the surface of the lake. It turns out that the energy of a wave traveling along the surface of the lake and the energy of a wave traveling at an angle to the surface of the lake add up. Therefore, in the evening the sound travels well across the lake. Even a quiet conversation can be heard standing on the opposite bank.

14. Huygens' principle– every point on the surface that the wave has reached at a given moment is a source of secondary waves. Drawing a surface tangent to the fronts of all secondary waves, we obtain the wave front at the next moment in time.

Let us consider, for example, a wave propagating along the surface of water from a point ABOUT(Fig.93) Let at the moment of time t the front had the shape of a circle of radius R centered at a point ABOUT. At the next moment of time, each secondary wave will have a front in the shape of a circle of radius, where V– speed of wave propagation. Drawing a surface tangent to the fronts of secondary waves, we obtain the wave front at the moment of time (Fig. 93)

If a wave propagates in a continuous medium, then the wave front is a sphere.

15. Reflection and refraction of waves. When a wave falls on the interface between two different media, each point of this surface, according to Huygens' principle, becomes a source of secondary waves propagating on both sides of the surface. Therefore, when crossing the interface between two media, the wave is partially reflected and partially passes through this surface. Because Because the media are different, the speed of the waves in them is different. Therefore, when crossing the interface between two media, the direction of propagation of the wave changes, i.e. wave refraction occurs. Let us consider, on the basis of Huygens' principle, the process and laws of reflection and refraction.

16. Law of Wave Reflection. Let a plane wave fall on a flat interface between two different media. Let us select the area between the two rays and (Fig. 94)

Angle of incidence - the angle between the incident beam and the perpendicular to the interface at the point of incidence.

Reflection angle is the angle between the reflected ray and the perpendicular to the interface at the point of incidence.

At the moment when the beam reaches the interface at point , this point will become a source of secondary waves. The wave front at this moment is marked by a straight line segment AC(Fig.94). Consequently, at this moment the beam still has to travel the path to the interface NE. Let the ray travel this path in time. The incident and reflected rays propagate on one side of the interface, so their velocities are the same and equal V. Then .

During the time the secondary wave from the point A will go the way. Hence . Right triangles are equal because... - common hypotenuse and legs. From the equality of triangles follows the equality of angles. But also, i.e. .

Now let us formulate the law of wave reflection: incident beam, reflected beam , perpendicular to the interface between two media, restored at the point of incidence, they lie in the same plane; the angle of incidence is equal to the angle of reflection.

17. Law of wave refraction. Let a plane wave pass through a flat interface between two media. Moreover the angle of incidence is different from zero (Fig. 95).

The angle of refraction is the angle between the refracted ray and the perpendicular to the interface, restored at the point of incidence.

Let us also denote the speed of propagation of waves in media 1 and 2. At the moment when the beam reaches the interface at the point A, this point will become a source of waves propagating in the second medium - a ray, and the ray still has to travel its way to the surface of the surface. Let be the time it takes the ray to travel NE, Then . During the same time, in the second medium the ray will travel the path . Because , then and .

Triangles and rectangles with a common hypotenuse, and =, are like angles with mutually perpendicular sides. For angles and we write the following equalities

Considering that , , we get

Now let us formulate the law of wave refraction: The incident ray, the refracted ray and the perpendicular to the interface between the two media, restored at the point of incidence, lie in the same plane; the ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant value for two given media and is called the relative refractive index for two given media.

18. Plane wave equation. Particles of the medium located at a distance S from the source of the waves begin to oscillate only when the wave reaches it. If V is the speed of wave propagation, then the oscillations will begin with a delay of time

If the source of waves oscillates according to a harmonic law, then for a particle located at a distance S from the source, we write the law of oscillations in the form

Let us introduce a quantity called the wave number. It shows how many wavelengths fit at a distance equal to units of length. Now the law of oscillations of a particle of a medium located at a distance S from the source we will write in the form

This equation determines the displacement of an oscillating point as a function of time and distance from the wave source and is called the plane wave equation.

19. Wave energy and intensity. Each particle that the wave reaches vibrates and therefore has energy. Let a wave with amplitude propagate in a certain volume of an elastic medium A and cyclic frequency. This means that the average vibration energy in this volume is equal to

Where m – mass of the allocated volume of the medium.

The average energy density (average over volume) is the wave energy per unit volume of the medium

Where is the density of the medium.

Wave intensity– a physical quantity numerically equal to the energy that a wave transfers per unit time through a unit area of a plane perpendicular to the direction of propagation of the wave (through a unit area of the wave front), i.e.

The average wave power is the average total energy transferred by the wave per unit time through a surface with area S. We obtain the average wave power by multiplying the wave intensity by the area S

20.The principle of superposition (overlay). If waves from two or more sources propagate in an elastic medium, then, as observations show, the waves pass through one another without affecting each other at all. In other words, the waves do not interact with each other. This is explained by the fact that within the limits of elastic deformation, compression and tension in one direction do not in any way affect the elastic properties in other directions.

Thus, every point in the medium where two or more waves arrive takes part in the oscillations caused by each wave. In this case, the resulting displacement of a particle of the medium at any time is equal to the geometric sum of the displacements caused by each of the resulting oscillatory processes. This is the essence of the principle of superposition or superposition of vibrations.

The result of the addition of oscillations depends on the amplitude, frequency and phase difference of the resulting oscillatory processes.

21. Coherent oscillations – oscillations with the same frequency and constant phase difference over time.

22.Coherent waves– waves of the same frequency or the same wavelength, the phase difference of which at a given point in space remains constant in time.

23.Wave interference– the phenomenon of an increase or decrease in the amplitude of the resulting wave when two or more coherent waves are superimposed.

A) . Interference maximum conditions. Let waves from two coherent sources meet at a point A(Fig.96).

Displacements of medium particles at a point A, caused by each wave separately, we will write according to the wave equation in the form

Where and , , are the amplitudes and phases of oscillations caused by waves at a point A, and are the distances of the point, is the difference between these distances or the difference in the wave paths.

Due to the difference in the course of the waves, the second wave is delayed compared to the first. This means that the phase of oscillations in the first wave is ahead of the phase of oscillations in the second wave, i.e. . Their phase difference remains constant over time.

In order to get to the point A particles oscillate with maximum amplitude, the crests of both waves or their troughs must reach the point A simultaneously in the same phases or with a phase difference equal to , where n – an integer, and - is the period of the sine and cosine functions,

Here, therefore, we write the condition of the interference maximum in the form

Where is an integer.

So, when coherent waves are superimposed, the amplitude of the resulting oscillation is maximum if the difference in the wave paths is equal to an integer number of wavelengths.

b) Interference minimum condition. Amplitude of the resulting oscillation at a point A is minimal if the crest and trough of two coherent waves simultaneously arrive at this point. This means that one hundred waves will arrive at this point in antiphase, i.e. their phase difference is equal to or , where is an integer.

We obtain the interference minimum condition by carrying out algebraic transformations:

Thus, the amplitude of oscillations when two coherent waves are superimposed is minimal if the difference in the wave paths is equal to an odd number of half-waves.

24. Interference and the law of conservation of energy. When waves interfere in places of interference minima, the energy of the resulting oscillations is less than the energy of the interfering waves. But in the places of interference maxima, the energy of the resulting oscillations exceeds the sum of the energies of the interfering waves to the extent that the energy in the places of interference minima has decreased.

When waves interfere, the oscillation energy is redistributed in space, but the conservation law is strictly observed.

25.Wave diffraction– the phenomenon of a wave bending around an obstacle, i.e. deviation from straight-line wave propagation.

Diffraction is especially noticeable when the size of the obstacle is smaller than the wavelength or comparable to it. Let there be a screen with a hole in the path of propagation of a plane wave, the diameter of which is comparable to the wavelength (Fig. 97).

According to Huygens' principle, each point of the hole becomes a source of the same waves. The size of the hole is so small that all the sources of secondary waves are located so close to each other that they can all be considered one point - one source of secondary waves.

If an obstacle is placed in the path of the wave, the size of which is comparable to the wavelength, then the edges, according to Huygens’ principle, become a source of secondary waves. But the size of the obstruction is so small that its edges can be considered coincident, i.e. the obstacle itself is a point source of secondary waves (Fig. 97).

The phenomenon of diffraction is easily observed when waves propagate over the surface of water. When the wave reaches a thin, motionless rod, it becomes the source of the waves (Fig. 99).

25. Huygens-Fresnel principle. If the dimensions of the hole significantly exceed the wavelength, then the wave, passing through the hole, propagates in a straight line (Fig. 100).

If the size of the obstacle significantly exceeds the wavelength, then a shadow zone is formed behind the obstacle (Fig. 101). These experiments contradict Huygens' principle. The French physicist Fresnel supplemented Huygens' principle with the idea of the coherence of secondary waves. Each point at which a wave arrives becomes a source of the same waves, i.e. secondary coherent waves. Therefore, waves are absent only in those places in which the conditions for an interference minimum are satisfied for secondary waves.

26. Polarized wave– a transverse wave in which all particles oscillate in the same plane. If the free end of the cord oscillates in one plane, then a plane-polarized wave propagates along the cord. If the free end of the cord oscillates in different directions, then the wave propagating along the cord is not polarized. If an obstacle in the form of a narrow slit is placed in the path of an unpolarized wave, then after passing through the slit the wave becomes polarized, because the slot allows vibrations of the cord to pass along it.

If a second slit is placed in the path of a polarized wave parallel to the first, then the wave will freely pass through it (Fig. 102).

If the second slit is placed at right angles to the first, then the spread of the ox will stop. A device that selects vibrations occurring in one specific plane is called a polarizer (first slit). The device that determines the plane of polarization is called an analyzer.

27.Sound - This is the process of propagation of compression and rarefaction in an elastic medium, for example, in gas, liquid or metals. The propagation of compression and rarefaction occurs as a result of the collision of molecules.

28. Sound volume This is the force of a sound wave on the eardrum of the human ear, which is caused by sound pressure.

Sound pressure – This is the additional pressure that occurs in a gas or liquid when a sound wave propagates. Sound pressure depends on the amplitude of vibration of the sound source. If we make a tuning fork sound with a light blow, we get the same volume. But, if the tuning fork is hit harder, the amplitude of its vibrations will increase and it will sound louder. Thus, the loudness of the sound is determined by the amplitude of the vibration of the sound source, i.e. amplitude of sound pressure fluctuations.

29. Pitch of sound determined by the frequency of oscillations. The higher the frequency of the sound, the higher the tone.

Sound vibrations occurring according to the harmonic law are perceived as a musical tone. Usually sound is a complex sound, which is a collection of vibrations with similar frequencies.

The fundamental tone of a complex sound is the tone corresponding to the lowest frequency in the set of frequencies of a given sound. The tones corresponding to the other frequencies of a complex sound are called overtones.

30. Sound timbre. Sounds with the same fundamental tone differ in timbre, which is determined by a set of overtones.

Each person has his own unique timbre. Therefore, we can always distinguish the voice of one person from the voice of another person, even when their fundamental tones are the same.

31.Ultrasound. The human ear perceives sounds whose frequencies range from 20 Hz to 20,000 Hz.

Sounds with frequencies above 20,000 Hz are called ultrasounds. Ultrasounds travel in the form of narrow beams and are used in sonar and flaw detection. Ultrasound can be used to determine the depth of the seabed and detect defects in various parts.

For example, if the rail does not have cracks, then ultrasound emitted from one end of the rail, reflected from its other end, will give only one echo. If there are cracks, then ultrasound will be reflected from the cracks and the instruments will record several echoes. Ultrasound is used to detect submarines and schools of fish. The bat navigates in space using ultrasound.

32. Infrasound– sound with a frequency below 20Hz. These sounds are perceived by some animals. Their source is often vibrations of the earth's crust during earthquakes.

33. Doppler effect is the dependence of the frequency of the perceived wave on the movement of the source or receiver of the waves.

Let a boat rest on the surface of a lake and let the waves beat against its side with a certain frequency. If the boat starts moving against the direction of wave propagation, then the frequency of waves hitting the side of the boat will increase. Moreover, the higher the speed of the boat, the higher the frequency of waves hitting the side. Conversely, when the boat moves in the direction of wave propagation, the frequency of impacts will become less. These reasoning can be easily understood from Fig. 103.

The higher the speed of oncoming traffic, the less time is spent covering the distance between the two nearest ridges, i.e. the shorter the period of the wave and the greater the frequency of the wave relative to the boat.

If the observer is stationary, but the source of the waves is moving, then the frequency of the wave perceived by the observer depends on the movement of the source.

Let a heron walk across a shallow lake towards the observer. Every time she puts her foot in the water, waves spread out in circles from this place. And each time the distance between the first and last waves decreases, i.e. A larger number of ridges and depressions are laid at a shorter distance. Therefore, for a stationary observer in the direction towards which the heron is walking, the frequency increases. And vice versa, for a stationary observer located at a diametrically opposite point at a greater distance, there are the same number of crests and troughs. Therefore, for this observer the frequency decreases (Fig. 104).